It is well established in the literature of finance that the internal rate of return (IRR) of an investment is calculated by IRR=r where
            ∑              i        =        0            n        ⁢                  CF        i                              (                      1            +            r                    )                i              =  0
It is also common knowledge in the finance industry and literature that the discount rate for actual IRR (r) and the discount rate for pro forma IRR (rpf) are the same when all cash flows of an investment are multiplied by a constant k:
  r  =                    r        pf            ⁢                          ⁢      where      ⁢                          ⁢                        ∑                      i            =            0                    n                ⁢                              kCF            i                                              (                              1                +                                  r                  pf                                            )                        i                                =    0  
This is so because the relative weights of the cash flows are unchanged as a function of time when multiplied by a constant.
Another way to understand why multiplying each cash flow by a constant does not change the IRR of an investment is to look at the original investment as a bond and the IRR as its yield to maturity. It is obvious that buying two identical bonds at the same price on the same date and with the same cash flows (and thus the same yield to maturity) would result in a portfolio with the same yield to maturity as that of the underlying bonds. The same would be true of buying 4 bonds or k bonds. It is a small extension of the principle to apply the same notion to fractional bonds and thus to all the cash flows multiplied by any constant k.
Another technical definition of IRR is the discount rate required to make the positive cash flows (PCF) resulting from the investment equal to the negative cash flows (NCF) expended in acquiring the investment:
            ∑              i        =        0            n        ⁢                  NCF        i                              (                      1            +            r                    )                i              =            ∑              i        =        0            n        ⁢                  PCF        i                              (                      1            +            r                    )                i            
It is therefore mathematically obvious that
            ∑              i        =        0            n        ⁢                  kNCF        i                              (                      1            +            r                    )                i              =            ∑              i        =        0            n        ⁢                  kPCF        i                              (                      1            +            r                    )                i            
An alternative method of IRR computation is referred to in the industry as the time-zero method. In the time-zero IRR method, all investments are presumed to begin at the same date (the zero date). In a 1995 white paper entitled Opportunistic Investing: Performance Measurement, Benchmarking and Evaluation, Richards and Tierney, a well-known consulting firm, has argued that the time-zero method is the best way to determine stock selection ability, since it neutralizes the relative timings of the acquisitions in a private market portfolio.
In the public markets, time weighted rate of return (TWROR) performance attribution has been refined to enable the analyst to determine the relative contribution of the stock index, sector allocation and stock selection in order to derive the manager's contribution, as shown in the numerical example below: